λ$$ \lambda $$-spheres as a new reference model for the geoid

We propose here a new reference model approximating the actual shape of the Earth. It is based on lambda$$ \lambda $$-spheres as an alternative to the rotational ellipsoids. Relying on quite convenient parametrization of the lambda$$ \lambda $$-spheres, we have obtained the concise expression for its z$$ z $$-coordinate as a combination of the incomplete elliptic integrals of all three kinds. Next, we have derived the geodesic equations on the lambda$$ \lambda $$-spheres and solved them in the analytical form using the mechanical method developed in our previous papers. As a result, three kinds of geodesics have been obtained for different values of the integration constant C2$$ {C}_2 $$, that is, the meridians, the Equator, and the slant geodesics. We have also found the conditions that should be fulfilled in order to obtain a closed slant geodesics, whereas the meridians and the Equator are closed by construction. Finally, we have presented the comparison of the newly proposed lambda$$ \lambda $$-sphere's model of the geoid with the most popular ellipsoidal reference models.

Automated summary

A new reference model approximating the actual shape of the Earth is proposed, based on lambda-spheres as an alternative to rotational ellipsoids. The z-coordinate of the lambda-spheres is expressed concisely using incomplete elliptic integrals. Geodesic equations on the lambda-spheres are derived and solved analytically, resulting in three types of geodesics. Conditions for obtaining closed slant geodesics are found, and a comparison with the popular ellipsoidal reference models is presented.